Bose-Hubbard models with on-site and nearest-neighbor interactions: Exactly solvable case

Abstract

We study the discrete spectrum of the two-particle Schrödinger operator H µλ (K), K ∈ T 2 , associated to the Bose-Hubbard Hamiltonian H µλ of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice Z 2 with interaction magnitudes µ ∈ R and λ ∈ R, respectively. We completely describe the spectrum of H µλ (0) and establish the optimal lower bound for the number of eigenvalues of H µλ (K) outside its essential spectrum for all values of K ∈ T 2 . Namely, we partition the (µ, λ)-plane such that in each connected component of the partition the number of bound states of H µλ (K) below or above its essential spectrum cannot be less than the corresponding number of bound states of H µλ (0) below or above its essential spectrum.